Mass Transfer in Packed Beds at Low Peclet Numbers

8/11/2019 Mass Transfer in Packed Beds at Low Peclet Numbers

1/9

P e r g a m o n

( h , 'm~ , . I t im/m , ' . ' rm, / S: w e . Vol 52 . Nos . 21 22. pp . 3995 41~)3 . I t~97

1 '4 q 7 | q s c ~ . lc r , ~ w n c e l . l d A l l r i g h t s r e s c r ' . e d

P r i n t e d i n G r e a t B r i t a i n

Plh SO009-25A)9(97)OO242-X ~ ~ ~ s l ~ l ~ -~l~)

Mass t rans fer in p ac k ed b ed s a t l ow Pec l e t

n u m b e r s w r o n g e x p e r i m e n t s o r w r o n g

interpretat ions?

G. Rexwinkel ,* A. B. M.

Heesink and

W. P. M.

Van Swaaij

Department of Chemical Engineering, Twcnte Universi ty of Technology, PO Box 217. 7500 AE

Enschcdc, The Netherlands

(Accepted 7 July 1997)

Abstract Much research has been focused on mass transfer phenomena in packed beds. i-'or

Peclet numbers above 200, empirical relations have been derived that predict the valuc of the

mass transfer coefficient as a funct ion of the Reyno lds numb er and the Schmidt number . These

relations are more or less simila r to the well-known relation that Ranz and Marshall derived for

mass transfer aroun d a single sphere in an infinite medium

S h = ~ + f l R e S d .

For packed beds of spherical particles an :~-value of 3.89 can be calculated on basis of

fundame ntal considerations. However, Sherwood number s much lower than this minim um

value have been observed at Peclct numbers below 100. Several expl anat ions have been

proposed for this apparent discrepancy, such as misinterpr etation of the experimental results

due to unjustified neglection of axial dispersion or wall channeling. In this work, a model that

predicts the combined effects of axial dispersion and wall channel ing has been developed. With

this model, it is possible to explain the results obtained with undiluted beds in which all

particles are active in the process of mass transfer. However, such an explan atio n is not possible

for the results obta ined with dilu ted beds in which not all particles arc active. Therefore, in the

case of diluted beds other reasons for the apparen t drop in mass transfer rate must exist. In the

present investigation, it is demonstrat ed that the drop again originates from misinterpret ation

of the experimental results. It is shown, both experimentally and theoretically, that low

Sherwood number s can be obtained when large differences cxist between the local concent ra-

tion, experienced by an active particle and the mixed cup concentration of the whole bed

cross-section. (" 1997 Elsevier Science l,td

K e y w o r d s : Mass transfer; packed beds: low Peclet numbers; dilution.

I. INTRODUCTION

The rate of mass transfer between particles and fluid

in packed-bed contactors in often predicted with the

help of a generalized, dimensionless correlation

S h

= function

( R e , S c ,

Geomet ry . (11

Several correlation s have been published, e.g. by

Wakao and Funazkr i (1978) and Gnielinski (1978j.

Most of these were based on mass transfer measure-

ments performed at high Peclet numbers ( P e > 200),

and are modified versions of the well-known

Ranz- Mars hall equation, that was derived for a single

sphere in an infinite medium. Though these correla-

tions are accurate at high Peclet numbers, at low

*Corresponding aulhor. Te l. : 0031 534894338: filx:

0031534894774.

Peclet numbers ( P e < 100) Sherwood numbers have

been measured that are less than predicted. See Fig. 1.

in which the results of several studies are presented.

Sorcnscn and Stewart (1974) examined mass trans-

fer in packed beds on a fundamental basis. They

demonstrated that, at low Reynolds numbers

I R e < 10), the rate of mass transfer is only a function

of the Peclet number. They also showed that the

minimum value of the Sherwood number, which is

reached at

Pe = O,

amounts to 3.89 in the case of

ideally packed spherical particles. A similar result was

obta ined by Gu nn (1978). Up to now, this result could

not bc confirmed experimentally, not even when ap-

plying sh al lo t beds to minimize experimental inac-

curacies, that are inevitable due to two complications

that appear when operating at low Peeler numbers:

(i) the number of mass transfer units becomes quite

large, making it difficult to determine the drivi ng force

for mass transfer in an accurate way, and (i i) the

3995


2/9

3996 G. Rexwinkel t a l

1 0 0 0 ~

100 ~ ~-

0

. . . . . . .

"

0.1

0.01

0.1 1 10 100 1000 10 00 0 100000

Williamson t al (1963)

Wilson nd Geankoplis 1966)

Appel nd Newman(1976)

Fedkiwand Newman(1982)

o Hsiung nd Thodos (1977)

13 Hobson nd Thodos (1951)

O Satterfield nd Resnick (1954)

Hsiung nd Thodos (1977)

Bar-Ilan nd Resnick(1957)

Wilson nd Geankop is (1966)

- - G n i e l i n s k i

Sc= 1000 (liquid)

....... GnielinskiSc=2.5 (gas)

1000000

P e [ ]

Fig. 1. Experimental Sherwood numbers from literature. Black symbols: undiluted bed--liquid; White

symbols; undiluted bed--gas; Gray symbols: diluted bed--gas (Bar-llan and Resnick, 1957; Hsiung and

Thodos, 1977) and diluted bed--liquid (Wilson and Geankoplis, 1966).

cont ribu tion of free convection to total mass transfer

may become significant. In order to overcome the first

complication some researchers applied diluted beds

in which only a part of the particles is active in the

mass transfer process, e.g . Bar -lla n and Resnick

(1957), Wilson and Ge ankopli s (1966)and Hsi ung and

Thodos (1977). By doing so they could lower the

concen tratio n in the effluent and calculate the drivi ng

force for mass transfer more precisely. Nevertheless,

their results are still not in accordance with expecta-

tions on basis of theory, as is illustrated by Fig. 1. The

correlation proposed by Gnielinski is assumed to de-

scribe the actual Sherwood number satisfactorily for

the whole Peclet range, since it predicts a limiting

Sherwood number of 3.8 at zero flow, which is in

agreement with the theoretical findings of Sorensen

and Stewart (1974), while at higher Peclet numbers

this correlation has also proven to be accurate.

In this paper, we will demonstrate that the discrep-

ancy between measured and predicted Sherwood

numbers, which is observed at low Peclet numbers, is

caused by wrong calculation of the (local) driving

force for mass transfer. We will do so both for un-

diluted beds, in which all particles are active, and for

diluted beds, in which only part of the particles in

active. However , emphasi s will be on diluted beds. It

will be shown that such beds need a comple tely differ-

ent approach than und iluted beds, and that the results

obtained with diluted beds are of no value to those

interested in mass transfer phenomena in undiluted

beds, at low Peclet numbers.

2. RE-EVALUATIONOF EXPERIMENTALDATA

Most Sherwood numbers, that were reported in

literature, where calculated while assuming plug flow

behavior of the fluid flowing through the bed, i.e.

while neglecting the effects of axial dispersion and

wall channeling. This may not always be justified,

especially not at small Peclet numbers. For example,

in the case of shallow beds, axial dispersion may have

a significant negative effect on the average driving

force for mass transfer, causing an apparent drop in

Sherwood number as was illustrated by Wakao and

Funazkri (1978). In the case of small diameter beds,

wall channeling may result in overestimation of the

average driving force in the bed, also resulting in an

apparent drop in Sherwood number (Martin, 1978).

As most reported data were obtained from

measurement s that were carried out with shallow beds

with a small diameter, it makes sense to re-evaluate

these data while correcting for the influences of axial

dispersion and wall channeling. We did so with the

help of the model developed by Martin (1978), which

was expanded in the following ways:

Mass transfer is assumed also to occur in the wall

zone. The mass t rans fer coefficients in the wall

zone and in the center part of the bed are as-

sumed to be equal.

The fluid concentrations in the effluents from

both zones are calculated on the basis of the

axially dispersed plug-flow model, in a similar

way as was done by Wakao and Funa zkri (1978).

The gas dispersion coefficient is calculated with

a correlation proposed by Gunn (1987), whereas

the liquid dispersion coefficient was calcula ted

using a correlation of Chung and Wen (1968).

These correlations have been proven to be accu-

rate also at low Peclet numbers.

The results of the re-evaluation are shown in Fig. 2. It

appears that, in the case of undiluted beds, the re-

evaluated Sherwood numbers reasonably agree with


3/9

Mass transfer in packed beds 3997

1000

100

10

0.1

0.01

. . - ' ~ * " "

i,d J

[ ]

Ga - .11'

. t

O

0 . I 1 1 0

1 00 I (X X ) 1 0 0 0 0 1 0 0 ( O ) 1 0 0 00 0 0

P e [ - I

Williarnson et al (1%3)

W ilson and Geankopl is (1966)

A p p e l a n d Ne wma n ( 1 97 6 )

Fedkiw an d Newm an (1982)

H s i u n g a n d Th o d o s ( 1 97 7 )

Q H o b s o n a n d Th o d o s ( 1 95 1 )

o Sat ter f ie ld an d Resnick (1954)

# H s i u n g a n d Th o d o s ( 1 97 7 )

o Bar - l lan and Resnick (1957)

g W ilson an d Geankopl is (1966)

Gnielin ki Sc=1000 (l iquid)

. . . . . . . Gn i e li n sk i Sc = 2 .5 ( g a s )

Fig . 2. Re-evaluated S herwood num bers . Black symbols : und i lu ted bed - - l i qu id: Wh i te symbols : und i lu ted

b e d - - g a s , Gr a y s y mb o l s: d i l u te d b e d g a s iBa r - I l a n a n d R e s ni c k, 1 9 5 7 ; H s i u n g a n d Th o d o s , 1 9 7 7 ) a n d

di lu ted bed l iquid (Wilson and G eank opl is . 19661.

t h e o ry . A p p a r e n t l y , t h e d r o p i n S h e r w o o d n u m b e r .

t h a t i s o b s e r v e d i n u n d i l u t e d b e d s a t d e c r e a s i n g P e c l e t

n u m b e r , i s c a u s e d b y w r o n g i n t e r p r e t a t i o n o f t h e

e x p e r i m e n t a l r e s u lt s , i.e . b y u n j u s t i f i e d n e g l e c t i o n o f

a x i al d i s p e r s i o n a n d / o r w a l l c h a n n e l i n g .

R e - e v a l u a t i o n h a r d l y h a s a n e ffe ct o n t h e d a t a o b -

t a i n e d w i t h d i l u t e d b e d s . W e t h e r e f o r e c o n c l u d e t h a t ,

i n t h e c a s e o f d i l u te d b e d s , o t h e r p h e n o m e n a t h a n

a x i al d i s p e r s i o n o r w a l l c h a n n e l i n g c a u s e t h e d r o p i n

S h e r w o o d n u m b e r w h i c h i s o b s e r v e d a t d e c re a s i n g

P e c l e t n u m b e r . B e l o w , w e w i l l tr y t o i d e n t i f y th e s e

p h e n o m e n a . W e s t a r t w i t h a s t u d y o f a s i n g l e a c t i v e

p a r t i c l e s u r r o u n d e d b y i n a c t i v e p a r t i c l e s o n l y . T h e n ,

w e w i l l e x a m i n e b e d s c o n t a i n i n g m u l t i p le a c ti v e p a r -

t ic les .

3 . D I L l ? T E D B E D S : A S I N G L E A C T I V E P A R T I C I , E

3 .1 . Th em e t ica l

R a n z a n d M a r s h a l l ( 1 95 2 ) e x a m i n e d m a s s t r a n s fe r

a r o u n d a s i n g l e s p h e r e l o c a t e d i n a n i n f i n i t e m e d i u m ,

i n w h i c h n o o t h e r p a r t i c l e s a re p r e s e n t . T h e y s h o w e d

i n a s im p l e w a y t h a t t h e m i n i m u m v a l u e o f t h e S h e r -

w o o d n u m b e r (a t P e = 0 ) a m o u n t s t o 2 . I n t h e c a s e

t h a t t h e s p h e r e i s l o c a t e d i n a n i n f i n i t e b e d o f i n a c t i v e

p a r t ic l e s t h e v a lu e o f t h e m i n i m u m S h e r w o o d n u m b e r

i s b y d e f i n i t i o n o b t a i n e d b y m u l t i p l y i n g t h i s v a l u e o f

2 w i t h t h e p o r o s i t y ~: o f t h e b e d a n d b y d i v i d i n g i t b y

t h e t o r t u o s i t y r o f t h e b e d :

S h m i n

= - - 2 .

( 2 )

T

T h e t o r t u o s i t y o f a r a n d o m l y p a c k e d b e d c a n b e

c a l c u l a t e d a c c o r d i n g t o ( P u n c o c h a r a n d D r a h o s ,

1 9 9 3 ) :

r - (3)

y i e l d i n g , Sh,,~, = 2 t: 1 s . T h i s i s i n a g r e e m e n t w i t h t h e

r e s u l ts o b t a i n e d b y M i y a u c h i (1 97 1) . T h e m i n i m a l

v a l u e t h u s a m o u n t s t o a b o u t 0 . 5 w h i c h i s lo w e r t h a n

t h e m i n i m a l v a l u e p r e d i c t e d f o r u n d i l u t e d b e d s , i n

w h i c h t h e c o n c e n t r a t i o n p r o fi le a r o u n d a p a r ti c le d o e s

n o t r c a c h b e y o n d t h e n e i g h b o r i n g p a r t ic l e s .

3.2. Exp er imen ta l

S o l i d - g a s m a s s t r a n s f e r c o e f fi c i en t s o f a s i n g l e ac -

t iv e s p h e r e i n a n i n a c t i v e b e d h a v e b e e n d e t e r m i n e d

b y m e a s u r i n g t he e v a p o r a t i o n r a te o f a c a m p h o r

s p h e r e w i th a p u r i ty 9 6 % a n d a d i a m e t e r o f 1 c m i n

a n i t r o g e n g a s s t re a m . A c a m p h o r s p h e r e w a s w e i g h e d

b e fo r e t h e e x p e r i m e n t a n d s u b s e q u e n t l y p l a c e d i n s i d e

t h e c e n t e r o f a r e g u l a r l y p a c k e d b e d o f I c m d i a m e t e r

g l a s s s p h e r e s , h a v i n g a d i a m e t e r o f 1 0 . 5 c m a n d

a h e i g h t o f 1 2 c m [ F i g . 3 (a ) ] . B e f o re e n t e r i n g t h e

p a c k e d b e d , th e n i t r o g e n g a s w a s l e d t h r o u g h a d i s-

t r i b u t o r w i t h a h e i g h t o f 6 c m , c o n s i s t i n g o f g l a ss

b e a d s o f 1 m m , i n o r d e r t o e n s u r e a n e v e n l y d i s t r i b -

u t e d g a s f lo w . A f t e r a c e r t a i n t i m e o f o p e r a t i o n , d u r i n g

w h i c h t h e g a s f l o w w a s d i r e c t e d e i t h e r u p f l o w o r

d o w n f l o w t h r o u g h t h e b e d , t h e c a m p h o r s p h e r e w a s

w e i g h e d a g ai n . F r o m t h e c h a n g e i n m a s s t h e m a s s

t r a n s f e r c o e f f i c i e n t w a s c a l c u l a t e d :

k = m o - m . . . . ( 4 )

t ~ . ~ x d 2 ( C , - C b ) M

I n a f i r s t e x p e r i m e n t a c a m p h o r s p h e r e w a s w e i g h e d ,

p l a c e d i n s i d e t h e p a c k e d b e d a n d a f t e r a w h i l e , n o t

a p p l y i n g a n i t r o g e n g a s s t r e a m , t a k e n o u t o f t h e b e d


4/9

3998 G. Rexwinkel t a l .

Sin~l ~

active

p r t i c l e

Vent ]

--L

12 cm

Distributor

10.5 cm

~ UV-meter J

Inert layer of

glass beads

Inert layer of

glass beads

5 c m

i

2 5m C m

).

10.5 cm

a) b)

Fig. 3. Schematicof the experimental setup for two different ypes of experimens: a) Single active sphere in

an inactive packed bed of glass spheres; (b) Multiple active spheres in a bed of inactive glass spheres which

is located between two layers of inactive glass spheres.

Table 1. Physical properties (at 25~C) ogether with design data and operating conditions

applied during the experiments

Single ac ti ve Multiple active

particle particles

Transferred species Camphor Methylbenzoate

Fluid Nitrogen gas Water

Temperature 20-25 C 20- 25:C

Molecular mass of transferred species 0.15224 kg/mol 0.13615 g/mol

Diffusion coefficient of transferred species 6.2

1 0 - 6 m / s *

9.3 10- lo m2/s

Solubility of transferred species -- 2.15 g/l:

Vapor pressure of transferred species 30.2 Pa~ ---

Particle diameter 10 mm 0.63-0.71 mm

Bed porosity 0.40 + 0.01 0.40 + 0.01

Percentage active particles -- 0.35%

Sc 2.54 1099

Re 0.11-141 0.001-- I

*Yaws (1995); *Wilke and Chang (1955); tGetzen e t

a l . (1992); Presser (1972)

and weighed again. It appeared that the reduction of

mass during this experiment could be neglected com-

pared to the reduction observed after a normal experi-

ment of several hours. During all experiments thc

decrease in particle diameter was negligible whereas

the increase of the camphor concentration in the ni-

trogen stream was small. Therefore, the bulk concen-

tration, Cb, could be considered equal to zero. The

saturation concentration, Cs, was determined from

the ideal gas law and the camphor vapor pressure,

P r e p , that was calculated for the applied temperature,

which varied between 20 and 25C but remained

constant during a single experiment. With these data

and the data from Table 1 available, the Sherwood

numb er was calculated according to

S h - R T ( m o - m r , , , ) (5)

t c x p n d p P v a p M D

In Fig. 4, the experimentally obtained Sherwood

numbers are plotted versus the Peclet number. The


5/9

Mass transl~cr

solid line in this figure represents the correlation pro-

posed by Ranz and Marshall (1952). The upflow and

downflow experiments gave equal results indicating

that the influence of free convection on the mass

transfer rate can be neglected. Below a Peeler number

of 50 the presence of inert particles becomes notice-

able causing the measured Sherwood number of fall

below the value predicted by the Ran z Marshall

equation. At very low Peclet numbers the Sherwood

number indeed approaches the theoretical minimum

of approximately 0.5, which is lower than the minimal

value of 3.89 in undilu ted beds. We may thus conclude

that diluted beds cannot be used to examine mass

transfer phenomena in undiluted beds at low Peeler

numbers. We also conclude that the drop in Sher-

wood number to wdues below the limit of 0.5, as

observed with diluted beds. e.g. Bar-ilan and Rcsnick

(1953), must bc caused by interaction of the active

particles present in the diluted bed. Wc will examine

this possible interaction in the next section.

4. DILUTED BEt)S:

M I .

I .TIPI,E ACTIVE PARTICI.ES

4 1 Theoretical

As has been stated earlier, it is important to know

the local drivin g force for mass transfer in order to be

able to calculate the local mass transfer coefficient. In

1(~ )

I 0

- r , ~ f .

D

0.1

1

Single article u p f l o v .

O Smgteparticle downtlow

I~ Bar-Ilan and R~n ick (19571

Ranz-Mar~hall

e q u a t k m

.~x=-2.5

(~L~,)

0 . 0 1 L , , ,

O. I I 0 I ~ 100 0 1(X](30

Pe

[ I

Fig. 4. Experimental Sherwood numbers for a single active

particle together with the results of multiple acti~,e particles

of Bar-ilan and Resnick 11957).

in packed beds 3999

the case of a single active sphere in an infinite bed of

inert particles, the driving force for mass transfer

experienced by the active particle in exactly known

and trivial. Determination of the driving force experi-

enced by an individual active particle in a diluted

packed bed. in which more active particles are pres-

ent. is less trivial.

In Fig. 5, several hypothetical distributions of ac-

tive particles in a two-dimens iona l packed bed are

shown. The driving force experienced by the active

particles will be much different in each situation. In

situation 1 all active particles experience the same

driving force which has not been influenced by the

presence of any active particles upstream. In situation

It all active particles have been placed directly on top

of each other. The influence of particles upstream on

the driving force experienced by the particles dov


6/9

4000

thoroughly mixed with spherical glass particles of the

same size. The mixture, of which 0.35% of all particles

was active, was brought into the bed resulting in an

active layer of 2 cm high. This layer was enclosed by

two 5 cm thick layers of inert glass beads. See Fig.

3(b). The methylbenzoate concentr ation at the exit of

the packed bed was continu ously measured using an

UV-spectrophotometer. The meth ylbenzoate concen-

tration in the effluent remained constant during the

first hou r of operation, indicatin g that pore limitat ion

in the porous Amberlite XAD-2 particles could be

neglected. The Sherwood numbers were calculated by

assuming plug-flow behavior

S h - d 2 ~ v p P In ( C s - C o u t )

6 D i n , \ C s - ~ (6)

The results are shown in Fig. 6 and are similar to

those of Bar-Ilan and Resnick (1957): Much smaller

values are obtained than with a single active particle.

In order to confirm that these smaller Sherwood

numbe rs are found because of intera ction between the

different active particles in the bed, extra experiments

were performed. These were done at a fixed Peclet

numb er of 2.89 using 0.5 g of impregnated Amberlite

XAD-2 particles. The number of mass transfer units

was thus kept constant. The height of the bed, includ-

ing the inert layers at the bottom and the top, always

was 12 cm. In orde r to simulate situati on I of Fig. 5 all

active particles were positioned in a horizontal layer

which was enlosed by two layers of inert glass beds

having the same diameter of 0.66 mm as the active

particles. In this way about 20% of the bed cross-

section was covered with active particles. Situat ion III

of Fig. 5 was also simulated. First all active particles

were mixed with equally sized inert glass beads. The

obtained mixture was then dumped into a tube with

a diameter of 1.35 cm a nd a height of 8 cm, which was

placed in the center of the bed, which was already

filled with a 2 cm thick layer ofiner t glass beads. Next,

the annular space in between the tube and the bed

wall was filled with inert glass beads and the tube was

slowly removed. Finally, a 2 cm thick layer of inert

G. Rexwinkelet al .

glass beads was added to complete the bed. Addi-

tional to these experiments, tests were performed us-

ing diluted layers of varying thickness [accord ing to

the set-up shown in Fig. 3(b)]. By appl ying a consta nt

amo unt of impregnated Amberlite XAD-2 particles of

0.5 g, the degree of dilution was varied. The results of

the interact ion experiments were evaluated on basis of

eq. (61. The ca lculated Sherwood numbers are shown

in Fig. 7.

If the as sumption of a radia lly well-mixed fluid

would hold in all cases, all experiments should have

given the same result, since equal amounts of active

particles have been applied. However, the results

show a significant influence of the particle distribu-

tion, which is in conflict with this assumption. The

calculated Sherwood numbers appear to increase

when the fraction of active particles is decreased,

especially at high degrees of dilution. This is in ac-

cordance with expectations, since the average axial

distance between the active particles strongly depends

on the degree of dilution at high dilution degrees,

whereas this dependence is relatively small at lower

degrees of dilut ion. So, the extent of radial mixing will

increase with the degree of dilution, in particular at

high degrees of dilution. The results obtai ned with the

particle distributions resembling situations I and III

of Fig. 5 also confirm that interaction between active

particles is an important item in diluted beds. This

interaction, which results from radial concentration

profiles, is not accounted for when assuming plug-

flow behavior. We will illustrate this with the model

discussed below.

4.3. T h e m o d e l

The importance of radial concentration profiles

and the resulting inter action between different active

particles in the bed can also be demonstrated in a the-

oretical way. To do so, the diluted bed is modelled by

a two-dimensional network of interconnected con-

tinuously stirred t ank reactors (CSTR); see Fig. 8.

Each CSTR contains exactly one particle which is

1 0 0

10

0 . 1

0 . 0 1

0.1

, Y

S m g t e p a r t i c l e . Th i s w o r k

I I I

0

M u l t i p l e p a r t i c l e s . T h i s w o r k

B a t - I l a n a n d R e sn i c k ( 1 9 5 7 )

1 3 t i s i u n g a n d T h o d o s ( 1 9 7 7 )

Ranz-Mar~,ha l l equa t i on Sc .=2 .5 (gas)

I I I

I 1 0 1 0 0 1 0 0 0 1 0 0 0 0

Pe

[-]

0 . 6 '

0.5

0.4

i

, ~ 0 . 3

0 . 2

0.I

S i n g l e m o n o l a y e r r e s e m b l i n g

s i t u a t i o n I o f F i g . 5.

o

T u b e n ' ~ u r e m e n t r e s em b l i ng

s i t u a t i o n I n o f F i g . 5 .

. . . . . . . . . . . . . : : ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

o d ~ l u t e d b e d m e a s u r e m e n t s

0 . 0 1 0 . 0 2 0 . 0 3

f rac t ion of partc i l es ac t ive

[-]

0 . 0 4

Fig. 6. Experimental Sherwood numbers for single particle Fig. 7. Experimental Sherwood numbers for several distri-

and multiple particles in a diluted bed. butions of active particle over the bed. P e = 2.89;S c = 1099.


7/9

Mass transfer

Fig. 8. Schem atic representat ion of the interconnected

C STRs mod el where the grey dots represent act ive particles;

Flowdirect ion is downward,

n x

= number of part icles in

ho rizon tal direct ion and ny = number of particles in vert ical

direction.

e i t h e r a c t i v e o r i n a c t i v e . W h e t h e r a p a r t i c l e is a c t iv e

o r n o t i s r a n d o m l y d e t e r m i n e d . A s o m e w h a t s i m i l a r

a p p r o a c h w a s a p p l i e d b y v a n d e n B l e ek (1 96 9) , w h o

i n v e s t i g a t e d t h e e ff e c t o f d i l u t i o n o n t h e d e g r e e o f

c o n v e r s i o n i n f ix e d - b e d c a t a l y t i c r e a c t o r s .

M a s s t r a n s f e r b e t w e e n C S T R s t a k e s p l a c e in a x i a l

d i r e c t i o n t h r o u g h c o n v e c t i o n a n d a x i a l d i s p e r s i o n

a n d i n l a te r a l d i r e c t i o n b y d i sp e r s i o n . T h e a m o u n t o f

m a t e r i a l t r a n s f e rr e d b y d i s p e r s i o n b e tw e e n t w o n e i g h -

b o r i n g C S T R s i s c a l c u l a t e d a c c o r d i n g t o

N ~. j .~ . i ~ t = - D ~ C i . i+ t - C , . j ( 7a )

d ,

C i - l , j - C i . j

N c i ~ i - 1 .~ = D l~ ,

(7b/

t i p

A s t a t i o n a r y d i m e n s i o n l e s s m a s s b a l a n c e f o r C S T R i.

j y i e l d s

2 2 C 1

0 = ( - I B o , ~ , B - ~ , , x ) ~ J + (1 B-~a , )C~. j

1 1

+ zo ~ - c ' ' i ~ ~ + __Bol~--C~_ ~.j

1 . . r t S h

+ __Bo,~--C i + t

~

+ n,~ j ~: P e (A C i4) ' (8)

W i t h C ~ . j d e f i n e d a s C ~ . j / C . , , C ~ b e i n g t h e c o n c e n t r a -

t i o n a t t h e s u r f a c e o f a n a c t i v e p a r t i c l e w h i c h i s e i th e r

d i s s o l v i n g o r e v a p o r a t i n g . I n t h is e q u a t i o n n ~, c a n

a d a p t t h e v a l u e s 0 o r 1 a n d i n d i c a t e s w h e t h e r o r n o t

a n a c t iv e p a r t i c l e is p r e s e n t i n C S T R

i , j .

T h e C S T R s

in packe d beds 4001

t h a t a r e l o c a t e d a g a i n s t t h e p a c k e d - b e d w a l l a r e o n l y

c o n n e c t e d t o C S T R s i n t h e i n t e r i o r o f t h e b e d .

A n i m p o r t a n t t e r m i s A C ~ , ~ . w h i c h r e p r e s e n t s t h e

d r i v i n g f o r c e e x p e r i e n c e d b y a n a c t i v e p a r t i c l e in

C S T R i , j . I n t h e c a s e o f u n d i l u t e d b e d s t h i s d r i v i n g

f o rc e s h o u l d b y d e f i n i t io n b e b a s e d o n t h e m i x e d c u p

c o n c e n t r a t i o n o f th e w h o l e b e d c r o s s- s e c t io n . H o w -

e v e r , i t i s n o t t o b e e x p e c t e d t h a t t h e s a m e d e f i n i t i o n

h o l d s f o r d i l u te d b e d s b e c a u s e o f r a d i a l c o n c e n t r a t i o n

g r a d i e n t s. A s w e a r e i n t e r e st e d i n d e m o n s t r a t i n g t h e

e f f e c t s o f n e g l e c t i n g t h e s e r a d i a l c o n c e n t r a t i o n g r a d i -

e n t s , i n t h e p r e s e n t m o d e l t w o d e f i n i t i o n s o f d r i v i n g

f o r c e a r e a p p l i e d : ( ij m i x e d c u p d r i v i n g f o rc e , w h i c h is

c a l c u l a t e d f r om t h e m i x e d c u p c o n c e n t r a t i o n in

a h o r i z o n t a l r o w , a n d l i il C S T R d r i v i n g f o r c e , w h i c h i s

c a l c u l a t e d o n b a s i s o f t h e c o n c e n t r a t i o n w i t h i n t h e

C S T R i n w h i c h t h e a c t i v e p a r t i c l e is l o c a t e d . I n m a t h -

e m a t i c a l f o r m

A C , j - - I - ~ " '" ( 9 a )

t I t2 . (

A C cj - -- 1 - C~. j . (9b)

E q u a t i o n ( 8 ) h a s b e e n s o l v e d u s i n g b o t h e x p r e s s i o n s

f o r t h e d r i v i n g f o r c e b y t h e s u c c e s s i v e o v e r r e l a x a t i o n

m e t h o d . A s i t m a k e s n o s e n s e t o a p p l y e x p r e s s i o n s

t h a t w e r e d e r i v e d f or t h r e e - d i m e n s i o n a l b e d s a n d t h a t

p r e d i c t a f i n a l v a l u e o f S h e r w o o d a t P e = 0 . t h e S h e r -

w o o d n u m b e r w a s c a l c u l a t e d s o m e w h a t a r b i t r a r i l y

a c c o r d i n g t o

S h = 0.3 P e 1 3 . (10)

T h e v a l u e s o f B o t, , a n d B o ~ x w e r e c a l c u l a t e d f r o m

c o r r e l a t i o n s r e p o r t e d b y G u n n (1 98 7) . F i g u r e 9 s h o w s

t h e c a l c u l a t e d l a t e r a l c o n c e n t r a t i o n p r o f i l e s a t t h e e x i t

o f a p a c k e d b e d . c o n t a i n i n g 2 0 0 x 4 0 p a r t i c l e s w i t h

a d i a m e t e r o f 0. 5 m m o f w h i c h 3 0 0 a r e a c t i v e , u s i n g

t h e t w o d i f f e re n t d e f i n i t i o n s o f d r i v i n g f o r c e .

F i g u r e 9 c l e a r ly s h o w s t h a t l a r g e l a t e r a l c o n c e n t r a -

t io n g r a d i e n t s m a y e x i s t in d i l u t e d p a c k e d b e d s . F u r -

t h e r m o r e , i t s h o w s t h a t n e g l e c t i n g t h e s e l a t e r a l

c o n c e n t r a t i o n g r a d i e n t s , w h i c h i s d o n e w h e n c a l c u l a t -

i n g t h e l o c a l d r i v i n g f o r c e fo r m a s s t r a n s f e r a c c o r d i n g

t o e q . (9 a l , c a n r e s u l t in l o c a l d i m e n s i o n l e s s c o n c e n t r a -

t i o n s w h i c h a r e l a r g e r t h a n u n i t y , w h i c h b y d e f i n i t i o n

i s i m p o s s i b l e . T h e l o c a l d r i v i n g f o r c e f o r m a s s t r a n s f e r

i n a d i l u t e d b e d i s t h u s o v e r e s t i m a t e d w h e n t h e r a d i a l

c o n c e n t r a t i o n p r o t i l e in t h e b e d i s n e g l e c t e d , a s i s

d o n e w h e n t h e p l u g - f lo w m o d e l is a p p l i e d t o e v a l u a t e

t h e e x p e r i m e n t a l r e s u l t s . T h i s i s a l s o i l l u s t r a t e d b y

F i g . 1 0 i n w h i c h t h c p r c d i c t e d S h e r w o o d n u m b e r i s

s h o w n a s a f u n c t io n o f th e P e c l e t n u m b e r f o r b o t h

d e f i n i t i o n s o f t h e l o c a l d r i v i n g f o r c e A C i 4 . T h e p r e -

s e n t ed S h e r w o o d n u m b e r s w e r e c a l c u l a t e d w h i l e

a s s u m i n g p l u g - f l o w b e h a v i o r o f t h e f lu i d , w h i c h i s

c o m m o n p r a c t i c e w h e n e v a l u a t i n g m a s s t r a n s f e r

e x p e r i m e n t s

I1X c; ~ ~ i 1 i .n~ ,

S h - P e l n I . I11)

t / a t : i t , c 7 C ~ . ~

o r


8/9

4O02

1 . 6

1 . 4

.2

1

0 . 6

0 . 4

0.2

0

G . R e x w i n k e l

et al .

I

M a J i m u m c o r v ~ m l rl U o n I ~ S S t bl

& ~ r u ~ o ~ l l l l ~ o n B I [ h e (

~ i t o f t l ~ I~ d

2 0 4 0 6 0 8 0 1 0 0

: : t , _ . . . . . . . o . _ , . , e

0.2

A l t l ' l g e Ol l t ' l l l l l l i o i l l [

~ t o t a , . I , ~

0

2 0 4 0 6 0 8 0 1 0 0

la teral positio n [ram] lateral pos ition [ram]

F ig . 9 . C a l c u l a t e d r a d i a l c o n c e n t r a t i o n p r o f i l e s a t t h e e x i t o f t h e p a c k e d b e d : ( a) M i x e d u p d r i v i n g f o rc e :

(b ) C STR dr iv i ng fo rce .

P e

= 10 : Sh = 1 .5 ; Bo i l , = 10 ; Bo~ , = 4 : d r = 0 .5 mm ; nx = 200 : n y = 40 :

z~

= 3 00: ~: = 0.48 = 1 - n:6.

10

s

,ll

0.1

O M xed cup driving orce

CSTRdriving orce

1 1 1

Pe 1-1

F ig . 1 0. S h e r w o o d n u m b e r s c a l c u l a t e d u s i n g e q . ( 1 1 ) f r o m

t h e m i x e d c u p c o n c e n t r a t i o n a t t h e e x i t o f t h e b e d a s c a l -

c u l a t e d b y t h e i n t e r c o n n e c t e d C S T R m o d e l u s i n g t h e t w o

d if fe ren t de f in i t ion s o f the d r iv ing fo rce . S o = 1000;

dp =

0.5 mm ; nx = 2(X): ny = 40; n~,,,, = 300: ~: = 0.48.

5 . CONCLUSIONS

T h e d e v i a t i o n b e t w e e n p r e d i c t ed a n d r e p o r t e d

S h e r w o o d n u m b e r s i n p a c ke d b e d s a t lo w P e c l e t n u m -

b e r s is c a u s e d b y t h e w r o n g i n t e r p r e t a t i o n o f t h e

e x p e r i m e n t a l d a t a . I n t h e c a s e o f u n d i l u t e d b e d s c o r -

r e c t i o n s s h o u l d b e m a d e f o r a x i a l d i sp e r s i o n a n d w a l l

c h a n n e l i n g . T h e S h e r w o o d n u m b e r s w h e n o b t a i n e d

r e a s o n a b l y a g r e e w i t h t h e o r y. S u c h c o r r e c t i o n s d o n o t

s u ff i ce in t h e c a s e o f d i l u t e d b e d s . T h e m i n i m a l S h e r -

w o o d n u m b e r i n d i l u t e d b e d s is l o w e r t h a n i n u n -

d i l u t e d b e d s. F u r t h e r m o r e . i n d i l u t e d b e d s t h e

e x i s t e n ce o f r a d i a l c o n c e n t r a t i o n p r o f il e s s h o u l d b e

c o n s i d e r e d , w h i c h i s n o t p o s s i b l e w h e n t h e d i s t r i b u -

t i o n o f t h e a c t i v e p a rt i c l es i s r a n d o m a n d t h e r e f o r e n o t

e x a c t l y k n o w n . N e g l e c t i o n o f th e s e r a d i a l c o n c e n t r a -

t i o n p r o f i le s b y a p p l y i n g o f th e p l u g - f l o w m o d e l r e -

s u it s i n u n d e r e s t i m a t e d S h e r w o o d n u m b e r s .

T h e r e f o r e , d i l u t e d b e d s s h o u l d n o t b e u s e d t o e x a m i n e

m a s s t r a n s f e r p h e n o m e n a i n u n d i l u t e d b e d s a t l o w

P e c l e t n u m b e r s .

T h e v a l u e s o b t a i n e d w h e n c a l c u l a t i n g t h e d r i v i n g

f o rc e o n b a s i s o f t h e m i x e d c u p c o n c e n t r a t i o n a r e

a l m o s t e q u a l t o t h o s e p r e d i c t e d b y e q . ( 1 0) . ( O n l y t h e

v a l u e a t

P e

= 1 i s s o m e w h a t l e s s d u e t o a x i a l d i s p e r -

s i o n w h i c h i s i n c l u d e d i n t h e m o d e l .) T h e v a l u e s o b -

t a i n e d w h e n c a l c u l a t i n g t h e d r i v i n g f o rc e o n b a si s o f C

t h e C S T R - c o n c e n t r a t i o n a r e m u c h l ow e r . W e , t he r e -

f o re , c o n c l u d e t h a t t h e d e v i a t i o n be t w e e n m e a s u r e d C

a n d p r e d ic t e d S h e r w o o d n u m b e r , w h i c h i s o b s e r v e d i n

d i l u t e d b e d s a t l o w P e c l e t n u m b e r s , is , a t l e a s t p a r t l y , A C ~ .j

c a u s e d b y t h e n e g l e c t i o n o f r a d i a l c o n c e n t r a t i o n p r o -

f i le s i n s i d e t h e b e d . d r

D u e t o i t s s i m p l i c i t y , t h e p r e s e n t m o d e l is n o t s u it e d D

t o q u a n t i f y t h e e f fe c t s o f r a d i a l c o n c e n t r a t i o n p r o f i l e s

o n t h e a v e r a g e r a t e o f m a s s t r a n s f e r i n s i d e a d i l u t e d D ~x

b e d . F o r t h is . C F D s i m u l a t i o n s a r e n e c e s s ar y . D la t

A c k n o w l e d g e m e n t s

T h i s i n v e s t i g a t i o n w a s s u p p o r t e d b y t h e D u t c h M i n i s t r y o f

E c o n o m i c A ff ai rs . T h e a u t h o r s a c k n o w l e d g e J . G 6 r t e n a n d

J . M . M e e r d i n k f o r t h e i r a s s i s t a n c e i n t h e e x p e r i m e n t a l

work

NOTATION

c o n c e n t r a t i o n o f t r a n s f e r r e d s p ec i e s in t h e

f l ui d , m o l / m 3

d i m e n s i o n l e s s c o n c e n t r a t i o n

C / C ~

d i m e n -

s i o n l e s s

d r i v i n g f o r ce e x p e r i e n c e d b y a n a c t i v e p a r -

t ic l e in C S T R

i . j

d i m e n s i o n l e s s

p a r t i c l e d i a m e t e r , m

d i f f u s i o n c o e f f i c i e n t o f t r a n s f e r r e d s p e c i e s ,

m 2 / s

a x i a l d i s p e r s i o n c o e f f i c i e n t , m Z / s

l a t e r a l d i s p e r s i o n c o e f f i c i e n t , m Z / s


9/9

k

m

M

n ~ J

Ilactivc

n x

n y

N

Pv ap

R

tcxo

T

r

m a s s t r a n s f e r c o e f f i c i e n t, m / s

m a s s , k g

m o l e c u l a r m a s s , k g / m o l

r a n d o m f a c t o r (0 o r 1 ) o f e q . 8 , d i m e n s i o n l e s s

n u m b e r o f a c t iv e p a r t i c l e s, d im e n s i o n l e s s

n u m b e r o f p a r t ic l e s in h o r i z o n t a l d i r e c t io n ,

d i m e n s i o n l e s s

n u m b e r o f p a r t i c l e s i n v e r t i c a l d i r e c t i o n , d i -

m e n s i o n l e s s

f lu x , m o l / m 2 s

v a p o r p r e s s u re o f c a m p h o r , P a

g a s c o n s ta n t , J / m o l K

d u r a t i o n o f e x p e ri m e n t , s

t e m p e r a t u r e , K

i n t e r s t i t i a l f l u id v e l o c i t y , m / s

Greek letter.s

~: b e d p o r o s i t y , d i m e n s i o n l e s s

r / v i s c os i t y , Pa s

v fluid flOW, m3."s

p d e n s i t y , k g / m 3

r t o r t u o s i t y , d i m e n s i o n l e s s

Subscripts

m i n m i n i m a l

0 i n i t i a l l y

h i n t h e b u l k

f w i t h r e s pe c t t o f l u i d

i r a d i a l p o s i t i o n o f C S T R

i n a t t he i n l e t

j a x i a l p o s i t i o n o f C S T R

o u t a t t h e o u t l e t

p w i t h r e s p e c t t o p a r t i c l e ( s )

s s a t u r a t i o n

Dimensionless numbers

oax

BOla

Sh

P e

R e

Sc

a x i a l B o d e n s t e i n n u m b e r ( = vdp/D~d

l a t e r a l B o d e n s t e i n n u m b e r ( = vdp/D~,,)

S h e r w o o d n u m b e r ( =

kdp/D)

i n t e r s ti t ia l P e c l et n u m b e r ( = vdp/D)

R e y n o l d s n u m b e r ( = plvdp/qy)

S c h m i d t n u m b e r ( = q//pyD)

R E F E R E N C E S

A p p e l , P . W . a n d N e w m a n , J . ( 19 7 6) A p p l i c a t i o n o f

t h e l i m i ti n g c u r re n t m e t h o d t o m a s s t r a n s f e r i n

p a c k e d b e d s a t v e r y l o w R e y n o l d s n u m b e r s .

A. I .Ch.E.J . 22 , 979 -984 .

B a r - I l a n , M . a n d R e s n i c k , W . ( 19 5 7) G a s p h a s e m a s s

t r a n s f e r i n fi x e d b e d s a t l o w R e y n o l d s n u m b e r s . Ind.

Engng Chem. 4 9 , 3 1 3 - 3 2 0 .

B l e e k v a n d e n ,

C. M.,

W i e l e v a n d e r , K . a n d B e r g v a n

d e n , P . J . (1 9 69 ) T h e e f f e ct o f d i l u t i o n o n t h e d e g r e e

o f c o n v e r s i o n i n f i x e d b e d c a t a l y t i c r e a c t o r s . Chem.

Engnq Sci. 24 , 681 -694 .

Mass transfer in packed beds 4003

C h u n g , S . F . a n d W e n , C . Y . ( 19 6 8) L o n g i t u d i n a l

d i s p e r s i o n o f l i q u i d f l o w i n g t h r o u g h f i x e d a n d

f l u i d i z e d be ds . A. I .Ch.E.J . 14 , 857 -866 .

F e d k i w , P . S . a n d N e w m a n , J . ( 19 8 2) M a s s - t r a n s f e r

c o e f f ic i e n ts i n p a c k e d b e d s a t v e r y l o w R e y n o l d s

n u m b e r . Int. d. Heat Mass Transfer 25 , 935 -943 .

G e t z e n , F . , H e f t e r, G . a n d M a c z y n s k i , A . ( 19 9 2) Solu-

bility Data Series, V o l . 4 8 . P e r g a m o n P r e s s , O x f o r d .

G n i e l i n s k i , V . ( 19 7 8) G l e i c h u n g e n z u r B e r e c h n u n g d e s

W ~ i r m e - u n d S t o f f a u s t a u c h e s i n d u r c h s t r 6 m t e n

r u h e n d e n K u g e l s c h i . i t t u n g e n b e i m i t t l e r e n u n d

g r o s s e n P e c l e t z a h l e n . VT-verfahrenstechnik 12,

363--366.

G u n n , D . J . ( 19 7 8) T r a n s f e r o f h e a t o r m a s s t o p a r -

t i c l e s i n f i xe d a nd f l u i d i s e d be ds . Int . . I . Heat Mass

TransJer 2 1 , 4 6 7 4 7 6 .

G u n n , D . J . (1 9 87 ) A x i a l a n d r a d i a l d i s p e r s i o n i n fi x e d

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